SFO: Learning PDE Operators via Spectral Filtering
cs.LG, cs.AI
Submitted: 2026-01-23
Updated: 2026-09-05
License: http://creativecommons.org/licenses/by/4.0/
The gist: Partial differential equations (PDEs) govern complex systems, yet neural operators often struggle to efficiently capture the long-range, nonlocal interactions inherent in their solution maps.
Terminology
Abstract
Partial differential equations (PDEs) govern complex systems, yet neural operators often struggle to efficiently capture the long-range, nonlocal interactions inherent in their solution maps. We introduce Spectral Filtering Operator (SFO), a neural operator that parameterizes integral kernels using the Universal Spectral Basis (USB), a fixed, global orthonormal basis derived from the eigenmodes of the Hilbert matrix in spectral filtering theory. Motivated by our theoretical finding that the discrete Green's functions of shift-invariant PDE discretizations exhibit spatial Linear Dynamical System (LDS) structure, we prove that these kernels admit compact approximations in the USB. By learning only the spectral coefficients of rapidly decaying eigenvalues, SFO achieves a highly efficient representation. Across six benchmarks, including reaction-diffusion, fluid dynamics, and 3D electromagnetics, SFO achieves state-of-the-art accuracy, reducing error by up to 40% relative to strong baselines while using substantially fewer parameters.
Sources
- LNO: Laplace Neural Operator for Solving Differential Equations
- Universal Learning of Nonlinear Dynamics
- Introduction to Online Control
- SVD-NO: Learning PDE Solution Operators with SVD Integral Kernels
- Neural Operator: Graph Kernel Network for Partial Differential Equations
- Flash STU: Fast Spectral Transform Units
- Transolver: A Fast Transformer Solver for PDEs on General Geometries
- Koopman neural operator as a mesh-free solver of non-linear partial differential equations
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